Method and apparatus for minimizing error in dynamic and steady-state processes for prediction, control, and optimization

ABSTRACT

A method for providing independent static and dynamic models in a prediction, control and optimization environment utilizes an independent static model ( 20 ) and an independent dynamic model ( 22 ). The static model ( 20 ) is a rigorous predictive model that is trained over a wide range of data, whereas the dynamic model ( 22 ) is trained over a narrow range of data. The gain K of the static model ( 20 ) is utilized to scale the gain k of the dynamic model ( 22 ). The forced dynamic portion of the model ( 22 ) referred to as the b i  variables are scaled by the ratio of the gains K and k. Thereafter, the difference between the new value input to the static model ( 20 ) and the prior steady-state value is utilized as an input to the dynamic model ( 22 ). The predicted dynamic output is then summed with the previous steady-state value to provide a predicted value Y.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.11/359,296, filed Feb. 21, 2006, entitled “Method and Apparatus forMinimizing Error in Dynamic and Steady-State Processes for Prediction,Control, and Optimization”, which is a Continuation in Part of U.S. Pat.No. 7,050,866, issued May 23, 2006, entitled “Dynamic Controller forControlling a System”, which is a Continuation of U.S. Pat. No.6,738,677, issued May 18, 2004, entitled “Method and Apparatus forModeling Dynamic and Steady-State Processes for Prediction, Control, andOptimization”, which is a Continuation of U.S. Pat. No. 6,487,459,issued Nov. 26, 2002, entitled “Method and Apparatus for ModelingDynamic and Steady-State Processes for Prediction, Control, andOptimization”, which is a Continuation of U.S. Pat. No. 5,933,345,issued Aug. 3, 1999, entitled “Method and Apparatus for Dynamic andSteady State Modeling Over a Desired Path Between Two End Points”.

TECHNICAL FIELD OF THE INVENTION

The present invention pertains in general to modeling techniques and,more particularly, to combining steady-state and dynamic models for thepurpose of prediction, control and optimization.

BACKGROUND OF THE INVENTION

Process models that are utilized for prediction, control andoptimization can be divided into two general categories, steady-statemodels and dynamic models. In each case the model is a mathematicalconstruct that characterizes the process, and process measurements areutilized to parameterize or fit the model so that it replicates thebehavior of the process. The mathematical model can then be implementedin a simulator for prediction or inverted by an optimization algorithmfor control or optimization.

Steady-state or static models are utilized in modern process controlsystems that usually store a great deal of data, this data typicallycontaining steady-state information at many different operatingconditions. The steady-state information is utilized to train anon-linear model wherein the process input variables are represented bythe vector U that is processed through the model to output the dependentvariable Y. The non-linear model is a steady-state phenomenological orempirical model developed utilizing several ordered pairs (U_(i), Y_(i))of data from different measured steady states. If a model is representedas:Y=P(U,Y)  (1)where P is some parameterization, then the steady-state modelingprocedure can be presented as:({right arrow over (U)},{right arrow over (Y)})→P  (2)where U and Y are vectors containing the U_(i), Y_(i) pair elements.Given the model P, then the steady-state process gain can be calculatedas:

$\begin{matrix}{K = \frac{\Delta\;{P\left( {U,Y} \right)}}{\Delta\; U}} & (3)\end{matrix}$The steady-state model therefore represents the process measurementsthat are taken when the system is in a “static” mode. These measurementsdo not account for the perturbations that exist when changing from onesteady-state condition to another steady-state condition. This isreferred to as the dynamic part of a model.

A dynamic model is typically a linear model and is obtained from processmeasurements which are not steady-state measurements; rather, these arethe data obtained when the process is moved from one steady-statecondition to another steady-state condition. This procedure is where aprocess input or manipulated variable u(t) is input to a process with aprocess output or controlled variable y(t) being output and measured.Again, ordered pairs of measured data (u(I), y(I)) can be utilized toparameterize a phenomenological or empirical model, this time the datacoming from non-steady-state operation. The dynamic model is representedas:y(t)=p(u(t),y(t))  (4)where p is some parameterization. Then the dynamic modeling procedurecan be represented as:({right arrow over (u)},{right arrow over (y)})→p  (5)Where u and y are vectors containing the (u(I),y(I)) ordered pairelements. Given the model p, then the steady-state gain of a dynamicmodel can be calculated as:

$\begin{matrix}{k = \frac{\Delta\;{p\left( {u,y} \right)}}{\Delta\; u}} & (6)\end{matrix}$Unfortunately, almost always the dynamic gain k does not equal thesteady-state gain K, since the steady-state gain is modeled on a muchlarger set of data, whereas the dynamic gain is defined around a set ofoperating conditions wherein an existing set of operating conditions aremildly perturbed. This results in a shortage of sufficient non-linearinformation in the dynamic data set in which non-linear information iscontained within the static model. Therefore, the gain of the system maynot be adequately modeled for an existing set of steady-state operatingconditions. Thus, when considering two independent models, one for thesteady-state model and one for the dynamic model, there is a mis-matchbetween the gains of the two models when used for prediction, controland optimization. The reason for this mis-match are that thesteady-state model is non-linear and the dynamic model is linear, suchthat the gain of the steady-state model changes depending on the processoperating point, with the gain of the linear model being fixed. Also,the data utilized to parameterize the dynamic model do not represent thecomplete operating range of the process, i.e., the dynamic data is onlyvalid in a narrow region. Further, the dynamic model represents theacceleration properties of the process (like inertia) whereas thesteady-state model represents the tradeoffs that determine the processfinal resting value (similar to the tradeoff between gravity and dragthat determines terminal velocity in free fall).

One technique for combining non-linear static models and linear dynamicmodels is referred to as the Hammerstein model. The Hammerstein model isbasically an input-output representation that is decomposed into twocoupled parts. This utilizes a set of intermediate variables that aredetermined by the static models which are then utilized to construct thedynamic model. These two models are not independent and are relativelycomplex to create.

SUMMARY OF THE INVENTION

The present invention disclosed and claimed herein comprises a methodand apparatus for controlling the operation of a plant by predicting achange in the dynamic input values to the plant to effect a change inthe output from a current output value at a first time to a desiredoutput value at a second time. The controller includes a dynamicpredictive model fore receiving the current input value and the desiredoutput value and predicting a plurality of input values at differenttime positions between the first time and the second time to define adynamic operation path of the plant between the current output value andthe desired output value at the second time. An optimizer then optimizesthe operation of the dynamic controller at each of the different timepositions from the first time to the second time in accordance with apredetermined optimization method that optimizes the objectives of thedynamic controller to achieve a desired path. This allows the objectivesof the dynamic predictive model to vary as a function of time.

In another aspect of the present invention, the dynamic model includes adynamic forward model operable to receive input values at each of thetime positions and map the received input values through a storedrepresentation of the plant to provide a predicted dynamic output value.An error generator then compares the predicted dynamic output value tothe desired output value and generates a primary error value as adifference therebetween for each of the time positions. An errorminimization device then determines a change in the input value tominimize the primary error value output by the error generator. Asummation device sums the determined input change value with theoriginal input value for each time position to provide a future inputvalue, with a controller controlling the operation of the errorminimization device and the optimizer. This minimizes the primary errorvalue in accordance with the predetermined optimization method.

In a yet another aspect of the present invention, the controller isoperable to control the summation device to iteratively minimize theprimary error value by storing the summed output value from thesummation device in a first pass through the error minimization deviceand then input the latch contents to the dynamic forward model insubsequent passes and for a plurality of subsequent passes. The outputof the error minimization device is then summed with the previouscontents of the latch, the latch containing the current value of theinput on the first pass through the dynamic forward model and the errorminimization device. The controller outputs the contents of the latch asthe input to the plant after the primary error value has been determinedto meet the objectives in accordance with the predetermined optimizationmethod.

In a further aspect of the present invention, a gain adjustment deviceis provided to adjust the gain of the linear model for substantially allof the time positions. This gain adjustment device includes a non-linearmodel for receiving an input value and mapping the received input valuethrough a stored representation of the plant to provide on the outputthereof a predicted output value, and having a non-linear gainassociated therewith. The linear model has parameters associatedtherewith that define the dynamic gain thereof with a parameteradjustment device then adjusting the parameters of the linear model as afunction of the gain of the non-linear model for at least one of thetime positions.

In yet a further aspect of the present invention, the gain adjustmentdevice further allows for approximation of the dynamic gain for aplurality of the time positions between the value of the dynamic gain atthe first time and the determined dynamic gain at one of the timepositions having the dynamic gain thereof determined by the parameteradjustment device. This one time position is the maximum of the timepositions at the second time.

In yet another aspect of the present invention, the error minimizationdevice includes a primary error modification device for modifying theprimary error to provide a modified error value. The error minimizationdevice optimizes the operation of the dynamic controller to minimize themodified error value in accordance with the predetermined optimizationmethod. The primary error is weighted as a function of time from thefirst time to the second time, with the weighting function decreasing asa function of time such that the primary error value is attenuated at arelatively high value proximate to the first time and attenuated at arelatively low level proximate to the second time.

In yet a further aspect of the present invention, a predictive system isprovided for predicting the operation of a plant with the predictivesystem having an input for receiving input value and an output forproviding a predicted output value. The system includes a non-linear,model having an input for receiving the input value and mapping itacross a stored learned representation of the plant to provide apredicted output. The non-linear model has an integrity associatedtherewith that is a function of a training operation that varies acrossthe mapped space. A first principles model is also provided forproviding a calculator representation of the plant. A domain analyzerdetermines when the input value falls within a region of the mappedspace having an integrity associated therewith that is less than apredetermined integrity threshold. A domain switching device is operableto switch operation between the non-linear model and the firstprinciples model as a function of the determined integrity levelcomparison with the predetermined threshold. If it is above theintegrity threshold, the non-linear model is utilized and, if it isbelow the integrity threshold, the first principles model is utilized.

BRIEF DESCRIPTION OF THE DRAWINGS

For a more complete understanding of the present invention and theadvantages thereof, reference is now made to the following descriptiontaken in conjunction with the accompanying Drawings in which:

FIG. 1 illustrates a prior art Hammerstein model;

FIG. 2 illustrates a block diagram of the modeling technique of thepresent invention;

FIG. 3a-3d illustrate timing diagrams for the various outputs of thesystem of FIG. 2;

FIG. 4 illustrates a detailed block diagram of the dynamic modelutilizing the identification method;

FIG. 5 illustrates a block diagram of the operation of the model of FIG.4;

FIG. 6 illustrates an example of the modeling technique of the presentinvention utilized in a control environment;

FIG. 7 illustrates a diagrammatic view of a change between twosteady-state values;

FIG. 8 illustrates a diagrammatic view of the approximation algorithmfor changes in the steady-state value;

FIG. 9 illustrates a block diagram of the dynamic model;

FIG. 10 illustrates a detail of the control network utilizing the errorconstraining algorithm of the present invention;

FIGS. 11a and 11b illustrate plots of the input and output duringoptimization;

FIG. 12 illustrates a plot depicting desired and predicted behavior;

FIG. 13 illustrates various plots for controlling a system to force thepredicted behavior to the desired behavior;

FIG. 14 illustrates a plot of the trajectory weighting algorithm of thepresent invention;

FIG. 15 illustrates a plot for the constraining algorithm;

FIG. 16 illustrates a plot of the error algorithm as a function of time;

FIG. 17 illustrates a flowchart depicting the statistical method forgenerating the filter and defining the end point for the constrainingalgorithm of FIG. 15;

FIG. 18 illustrates a diagrammatic view of the optimization process;

FIG. 18a illustrates a diagrammatic representation of the manner inwhich the path between steady-state values is mapped through the inputand output space;

FIG. 19 illustrates a flowchart for the optimization procedure;

FIG. 20 illustrates a diagrammatic view of the input space and the errorassociated therewith;

FIG. 21 illustrates a diagrammatic view of the confidence factor in theinput space;

FIG. 22 illustrates a block diagram of the method for utilizing acombination of a non-linear system and a first principal system; and

FIG. 23 illustrates an alternate embodiment of the embodiment of FIG.22.

DETAILED DESCRIPTION OF THE INVENTION

Referring now to FIG. 1, there is illustrated a diagrammatic view of aHammerstein model of the prior art. This is comprised of a non-linearstatic operator model 10 and a linear dynamic model 12, both disposed ina series configuration. The operation of this model is described in H.T. Su, and T. J. McAvoy, “Integration of Multilayer Perceptron Networksand Linear Dynamic Models: A Hammerstein Modeling Approach” to appear inI & EC Fundamentals, paper dated Jul. 7, 1992, which reference isincorporated herein by reference. Hammerstein models in general havebeen utilized in modeling non-linear systems for some time. Thestructure of the Hammerstein model illustrated in FIG. 1 utilizes thenon-linear static operator model 10 to transform the input U intointermediate variables H. The non-linear operator is usually representedby a finite polynomial expansion. However, this could utilize a neuralnetwork or any type of compatible modeling system. The linear dynamicoperator model 12 could utilize a discreet dynamic transfer functionrepresenting the dynamic relationship between the intermediate variableH and the output Y. For multiple input systems, the non-linear operatorcould utilize a multilayer neural network, whereas the linear operatorcould utilize a two layer neural network. A neural network for thestatic operator is generally well known and described in U.S. Pat. No.5,353,207, issued Oct. 4, 1994, and assigned to the present assignee,which is incorporated herein by reference. These type of networks aretypically referred to as a multilayer feed-forward network whichutilizes training in the form of back-propagation. This is typicallyperformed on a large set of training data. Once trained, the network hasweights associated therewith, which are stored in a separate database.

Once the steady-state model is obtained, one can then choose the outputvector from the hidden layer in the neural network as the intermediatevariable for the Hammerstein model. In order to determine the input forthe linear dynamic operator, u(t), it is necessary to scale the outputvector h(d) from the non-linear static operator model 10 for the mappingof the intermediate variable h(t) to the output variable of the dynamicmodel y(t), which is determined by the linear dynamic model.

During the development of a linear dynamic model to represent the lineardynamic operator, in the Hammerstein model, it is important that thesteady-state non-linearity remain the same. To achieve this goal, onemust train the dynamic model subject to a constraint so that thenon-linearity learned by the steady-state model remains unchanged afterthe training. This results in a dependency of the two models on eachother.

Referring now to FIG. 2, there is illustrated a block diagram of themodeling method of the present invention, which is referred to as asystematic modeling technique. The general concept of the systematicmodeling technique in the present invention results from the observationthat, while process gains (steady-state behavior) vary with U's and Y's,(i.e., the gains are non-linear), the process dynamics seemingly varywith time only, (i.e., they can be modeled as locally linear, buttime-varied). By utilizing non-linear models for the steady-statebehavior and linear models for the dynamic behavior, several practicaladvantages result. They are as follows:

-   -   1. Completely rigorous models can be utilized for the        steady-state part. This provides a credible basis for economic        optimization.    -   2. The linear models for the dynamic part can be updated        on-line, i.e., the dynamic parameters that are known to be        time-varying can be adapted slowly.    -   3. The gains of the dynamic models and the gains of the        steady-state models can be forced to be consistent (k=K).

With further reference to FIG. 2, there are provided a static.orsteady-state model 20 and a dynamic model 22. The static model 20, asdescribed above, is a rigorous model that is trained on a large set ofsteady-state data. The static model 20 will receive a process input Uand provide a predicted output Y. These are essentially steady-statevalues. The steady-state values at a given time are latched in variouslatches, an input latch 24 and an output latch 26. The latch 24 containsthe steady-state value of the input U_(ss), and the latch 26 containsthe steady-state output value Y_(ss). The dynamic model 22 is utilizedto predict the behavior of the plant when a change is made from asteady-state value of Y_(ss) to a new value Y. The dynamic model 22receives on the input the dynamic input value u and outputs a predicteddynamic value y. The value u is comprised of the difference between thenew value U and the steady-state value in the latch 24, U_(ss). This isderived from a subtraction circuit 30 which receives on the positiveinput thereof the output of the latch 24 and on the negative inputthereof the new value of U. This therefore represents the delta changefrom the steady-state. Similarly, on the output the predicted overalldynamic value will be the sum of the output value of the dynamic model,y, and the steady-state output value stored in the latch 26, Y_(ss).These two values are summed with a summing block 34 to provide apredicted output Y. The difference between the value output by thesumming junction 34 and the predicted value output by the static model20 is that the predicted value output by the summing junction 20accounts for the dynamic operation of the system during a change. Forexample, to process the input values that are in the input vector U bythe static model 20, the rigorous model, can take significantly moretime than running a relatively simple dynamic model. The method utilizedin the present invention is to force the gain of the dynamic model 22k_(d) to equal the gain K_(ss) of the static model 20.

In the static model 20, there is provided a storage block 36 whichcontains the static coefficients associated with the static model 20 andalso the associated gain value K_(ss). Similarly, the dynamic model 22has a storage area 38 that is operable to contain the dynamiccoefficients and the gain value k_(d). One of the important aspects ofthe present invention is a link block 40 that is operable to modify thecoefficients in the storage area 38 to force the value of k_(d) to beequal to the value of K_(ss). Additionally, there is an approximationblock 41 that allows approximation of the dynamic gain k_(d) between themodification updates.

Systematic Model

The linear dynamic model 22 can generally be represented by thefollowing equations:

$\begin{matrix}{{{\delta\;{y(t)}} = {{\sum\limits_{i = 1}^{n}\;{b_{i}\delta\;{u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}\;{a_{i}\delta\;{y\left( {t - i} \right)}}}}}{{where}\text{:}}} & (7) \\{{\delta\;{y(t)}} = {{y(t)} - Y_{ss}}} & (8) \\{{\delta\;{u(t)}} = {{u(t)} - u_{ss}}} & (9)\end{matrix}$and t is time, a_(i) and b_(i) are real numbers, d is a time delay, u(t)is an input and y(t) an output. The gain is represented by:

$\begin{matrix}{\frac{y(B)}{u(B)} = {k = \frac{\left( {\sum\limits_{i = 1}^{n}\;{b_{i}B^{i - 1}}} \right)B^{d}}{1 + {\sum\limits_{i = 1}^{n}\;{a_{i}B^{i - 1}}}}}} & (10)\end{matrix}$where B is the backward shift operator B(x(t))=x(t−1), t=time, the a_(i)and b_(i) are real numbers, I is the number of discreet time intervalsin the dead-time of the process, and n is the order of the model. Thisis a general representation of a linear dynamic model, as contained inGeorge E. P. Box and G. M. Jenkins, “TIME SERIES ANALYSIS forecastingand control”, Holden-Day, San Francisco, 1976, Section 10.2, Page 345.This reference is incorporated herein by reference.

The gain of this model can be calculated by setting the value of B equalto a value of “1”. The gain will then be defined by the followingequation:

$\begin{matrix}{\left\lbrack \frac{y(B)}{u(B)} \right\rbrack_{B = 1} = {k_{d} = \frac{\sum\limits_{i = 1}^{n}\; b_{i}}{1 + {\sum\limits_{i = 1}^{n}\; a_{i}}}}} & (11)\end{matrix}$

The a_(i) contain the dynamic signature of the process, its unforced,natural response characteristic. They are independent of the processgain. The b_(i) contain part of the dynamic signature of the process;however, they alone contain the result of the forced response. The b_(i)determine the gain k of the dynamic model. See: J. L. Shearer, A. T.Murphy, and H. H. Richardson, “Introduction to System Dynamics”,Addison-Wesley, Reading, Mass., 1967, Chapter 12. This reference isincorporated herein by reference.

Since the gain K_(ss) of the steady-state model is known, the gain k_(d)of the dynamic model can be forced to match the gain of the steady-statemodel by scaling the b_(i) parameters. The values of the static anddynamic gains are set equal with the value of b_(i) scaled by the ratioof the two gains:

$\begin{matrix}{\left( b_{i} \right)_{scaled} = {\left( b_{i} \right)_{old}\left( \frac{K_{ss}}{k_{d}} \right)}} & (12) \\{\left( b_{i} \right)_{scaled} = \frac{\left( b_{i} \right)_{old}{K_{ss}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (13)\end{matrix}$This makes the dynamic model consistent with its steady-statecounterpart. Therefore, each time the steady-state value changes, thiscorresponds to a gain K_(ss) of the steady-state model. This value canthen be utilized to update the gain k_(d) of the dynamic model and,therefore, compensate for the errors associated with the dynamic modelwherein the value of k_(d) is determined based on perturbations in theplant on a given set of operating conditions. Since all operatingconditions are not modeled, the step of varying the gain will accountfor changes in the steady-state starting points.

Referring now to FIGS. 3a-3d , there are illustrated plots of the systemoperating in response to a step function wherein the input value Uchanges from a value of 100 to a value of 110. In FIG. 3a , the value of100 is referred to as the previous steady-state value U_(ss). In FIG. 3b, the value of u varies from a value of 0 to a value of 10, thisrepresenting the delta between the steady-state value of U_(ss) to thelevel of 110, represented by reference numeral 42 in FIG. 3a .Therefore, in FIG. 3b the value of u will go from 0 at a level 44, to avalue of 10 at a level 46. In FIG. 3c , the output Y is represented ashaving a steady-state value Y_(ss) of 4 at a level 48. When the inputvalue U rises to the level 42 with a value of 110, the output value willrise. This is a predicted value. The predicted value which is the properoutput value is represented by a level 50, which level 50 is at a valueof 5. Since the steady-state value is at a value of 4, this means thatthe dynamic system must predict a difference of a value of 1. This isrepresented by FIG. 3d wherein the dynamic output value y varies from alevel 54 having a value of 0 to a level 56 having a value of 1.0.However, without the gain scaling, the dynamic model could, by way ofexample, predict a value for y of 1.5, represented by dashed level 58,if the steady-state values were outside of the range in which thedynamic model was trained. This would correspond to a value of 5.5 at alevel 60 in the plot of FIG. 3c . It can be seen that the dynamic modelmerely predicts the behavior of the plant from a starting point to astopping point, not taking into consideration the steady-state values.It assumes that the steady-state values are those that it was trainedupon. If the gain k_(d) were not scaled, then the dynamic model wouldassume that the steady-state values at the starting point were the samethat it was trained upon. However, the gain scaling link between thesteady-state model and the dynamic model allow the gain to be scaled andthe parameter b_(i) to be scaled such that the dynamic operation isscaled and a more accurate prediction is made which accounts for thedynamic properties of the system.

Referring now to FIG. 4, there is illustrated a block diagram of amethod for determining the parameters a_(i), b_(i). This is usuallyachieved through the use of an identification algorithm, which isconventional. This utilizes the (u(t),y(t)) pairs to obtain the a_(i)and b_(i) parameters. In the preferred embodiment, a recursiveidentification method is utilized where the a_(i) and b_(i) parametersare updated with each new (u_(i)(t),y_(i)(t)) pair. See: T Eykhoff,“System Identification”, John Wiley & Sons, New York, 1974, Pages 38 and39, et. seq., and H. Kurz and W. Godecke, “Digital Parameter-AdaptiveControl Processes with Unknown Dead Time”, Automatica, Vol. 17, No. 1,1981, pp. 245-252, which references are incorporated herein byreference.

In the technique of FIG. 4, the dynamic model 22 has the output thereofinput to a parameter-adaptive control algorithm block 60 which adjuststhe parameters in the coefficient storage block 38, which also receivesthe scaled values of k, b_(i). This is a system that is updated on aperiodic basis, as defined by timing block 62. The control algorithm 60utilizes both the input u and the output y for the purpose ofdetermining and updating the parameters in the storage area 38.

Referring now to FIG. 5, there is illustrated a block diagram of thepreferred method. The program is initiated in a block 68 and thenproceeds to a function block 70 to update the parameters a_(i), b_(i)the (u(I),y(I)) pairs. Once these are updated, the program flows to afunction block 72 wherein the steady-state gain factor K is received,and then to a function block 74 to set the dynamic gain to the steadystate gain, i.e., provide the scaling function described hereinabove.This is performed after the update. This procedure can be used foron-line identification, non-linear dynamic model prediction and adaptivecontrol.

Referring now to FIG. 6, there is illustrated a block diagram of oneapplication of the present invention utilizing a control environment. Aplant 78 is provided which receives input values u(t) and outputs anoutput vector y(t). The plant 78 also has measurable state variabless(t). A predictive model 80 is provided which receives the input valuesu(t) and the state variables s(t) in addition to the output value y(t).The steady-state model 80 is operable to output a predicted value ofboth y(t) and also of a future input value u(t+1). This constitutes asteady-state portion of the system. The predicted steady-state inputvalue is U_(ss) with the predicted steady-state output value being Y. Ina conventional control scenario, the steady-state model 80 would receiveas an external input a desired value of the output y^(d)(t) which is thedesired value that the overall control system seeks to achieve. This isachieved by controlling a distributed control system (DCS) 86 to producea desired input to the plant. This is referred to as u(t+1), a futurevalue. Without considering the dynamic response, the predictive model80, a steady-state model, will provide the steady-state values. However,when a change is desired, this change will effectively be viewed as a“step response”.

To facilitate the dynamic control aspect, a dynamic controller 82 isprovided which is operable to receive the input u(t), the output valuey(t) and also the steady-state values U_(ss) and Y_(ss) and generate theoutput u(t+1). The dynamic controller effectively generates the dynamicresponse between the changes, i.e., when the steady-state value changesfrom an initial steady-state value U_(ss) ^(i), Y_(ss) ^(i) to a finalsteady-state value U_(ss) ^(f), Y_(ss) ^(f).

During the operation of the system, the dynamic controller 82 isoperable in accordance with the embodiment of FIG. 2 to update thedynamic parameters of the dynamic controller 82 in a block 88 with again link block 90, which utilizes the value K_(ss) from a steady-stateparameter block in order to scale the parameters utilized by the dynamiccontroller 82, again in accordance with the above described method. Inthis manner, the control function can be realized. In addition, thedynamic controller 82 has the operation thereof optimized such that thepath traveled between the initial and final steady-state values isachieved with the use of the optimizer 83 in view of optimizerconstraints in a block 85. In general, the predicted model (steady-statemodel) 80 provides a control network function that is operable topredict the future input values. Without the dynamic controller 82, thisis a conventional control network which is generally described in U.S.Pat. No. 5,353,207, issued Oct. 4, 1994, to the present assignee, whichpatent is incorporated herein by reference.

Approximate Systematic Modeling

For the modeling techniques described thus far, consistency between thesteady-state and dynamic models is maintained by resealing the b_(i)parameters at each time step utilizing equation 13. If the systematicmodel is to be utilized in a Model Predictive Control (MPC) algorithm,maintaining consistency may be computationally expensive. These types ofalgorithms are described in C. E. Garcia, D. M. Prett and M. Morari.Model predictive control: theory and practice—a survey, Automatica,25:335-348, 1989; D. E. Seborg, T. F. Edgar, and D. A. Mellichamp.Process Dynamics and Control. John Wiley and Sons, New York, N.Y., 1989.These references are incorporated herein by reference. For example, ifthe dynamic gain k_(d) is computed from a neural network steady-statemodel, it would be necessary to execute the neural network module eachtime the model was iterated in the MPC algorithm. Due to the potentiallylarge number of model iterations for certain MPC problems, it could becomputationally expensive to maintain a consistent model. In this case,it would be better to use an approximate model which does not rely onenforcing consistencies at each iteration of the model.

Referring now to FIG. 7, there is illustrated a diagram for a changebetween steady state values. As illustrated, the steady-state model willmake a change from a steady-state value at a line 100 to a steady-statevalue at a line 102. A transition between the two steady-state valuescan result in unknown settings. The only way to insure that the settingsfor the dynamic model between the two steady-state values, an initialsteady-state value K_(ss) ^(i) and a final steady-state gain K_(ss)^(f), would be to utilize a step operation, wherein the dynamic gaink_(d) was adjusted at multiple positions during the change. However,this may be computationally expensive. As will be described hereinbelow,an approximation algorithm is utilized for approximating the dynamicbehavior between the two steady-state values utilizing a quadraticrelationship. This is defined as a behavior line 104, which is disposedbetween an envelope 106, which behavior line 104 will be describedhereinbelow.

Referring now to FIG. 8, there is illustrated a diagrammatic view of thesystem undergoing numerous changes in steady-state value as representedby a stepped line 108. The stepped line 108 is seen to vary from a firststeady-state value at a level 110 to a value at a level 112 and thendown to a value at a level 114, up to a value at a level 116 and thendown to a final value at a level 118. Each of these transitions canresult in unknown states. With the approximation algorithm that will bedescribed hereinbelow, it can be seen that, when a transition is madefrom level 110 to level 112, an approximation curve for the dynamicbehavior 120 is provided. When making a transition from level 114 tolevel 116, an approximation gain curve 124 is provided to approximatethe steady state gains between the two levels 114 and 116. For makingthe transition from level 116 to level 118, an approximation gain curve126 for the steady-state gain is provided. It can therefore be seen thatthe approximation curves 120-126 account for transitions betweensteady-state values that are determined by the network, it being notedthat these are approximations which primarily maintain the steady-stategain within some type of error envelope, the envelope 106 in FIG. 7.

The approximation is provided by the block 41 noted in FIG. 2 and can bedesigned upon a number of criteria, depending upon the problem that itwill be utilized to solve. The system in the preferred embodiment, whichis only one example, is designed to satisfy the following criteria:

-   -   1. Computational Complexity: The approximate systematic model        will be used in a Model Predictive Control algorithm, therefore,        it is required to have low computational complexity.    -   2. Localized Accuracy: The steady-state model is accurate in        localized regions. These regions represent the steady-state        operating regimes of the process. The steady-state model is        significantly less accurate outside these localized regions.    -   3. Final Steady-State: Given a steady-state set point change, an        optimization algorithm which uses the steady-state model will be        used to compute the steady-state inputs required to achieve the        set point. Because of item 2, it is assumed that the initial and        final steady-states associated with a set-point change are        located in regions accurately modeled by the steady-state model.

Given the noted criteria, an approximate systematic model can beconstructed by enforcing consistency of the steady-state and dynamicmodel at the initial and final steady-state associated with a set pointchange and utilizing a linear approximation at points in between the twosteady-states. This approximation guarantees that the model is accuratein regions where the steady-state model is well known and utilizes alinear approximation in regions where the steady-state model is known tobe less accurate. In addition, the resulting model has low computationalcomplexity. For purposes of this proof, Equation 13 is modified asfollows:

$\begin{matrix}{b_{i,{scaled}} = \frac{b_{i}{K_{ss}\left( {u\left( {t - d - 1} \right)} \right)}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}{\sum\limits_{i = 1}^{n}b_{i}}} & (14)\end{matrix}$

This new equation 14 utilizes K_(ss)(u(t−d−1)) instead of K_(ss)(u(t))as the consistent gain, resulting in a systematic model which is delayinvariant.

The approximate systematic model is based upon utilizing the gainsassociated with the initial and final steady-state values of a set-pointchange. The initial steady-state gain is denoted K^(i) _(ss) while theinitial steady-state input is given by U^(i) _(ss). The finalsteady-state gain is K^(f) _(ss) and the final input is U^(f) _(ss).Given these values, a linear approximation to the gain is given by:

$\begin{matrix}{{K_{ss}\left( {u(t)} \right)} = {K_{ss}^{i} + {\frac{K_{ss}^{f} - K_{ss}^{i}}{U_{ss}^{f} - U_{ss}^{i}}{\left( {{u(t)} - U_{ss}^{i}} \right).}}}} & (15)\end{matrix}$Substituting this approximation into Equation 13 and replacingu(t−d−1)−u^(i) by δu(t−d−1) yields:

$\begin{matrix}{{\overset{\sim}{b}}_{j,{scaled}} = {\frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}} + {\frac{1}{2}\frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}\delta\;{u\left( {t - d - i} \right)}}}} & (16)\end{matrix}$To simplify the expression, define the variable b_(j)-Bar as:

$\begin{matrix}{{\overset{\_}{b}}_{j} = \frac{b_{j}{K_{ss}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (17)\end{matrix}$and g_(j) as:

$\begin{matrix}{g_{j} = \frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}} & (18)\end{matrix}$Equation 16 may be written as:{tilde over (b)} _(j,scaled) = b _(j) +g _(j) δu(t−d−i).  (19)Finally, substituting the scaled b's back into the original differenceEquation 7, the following expression for the approximate systematicmodel is obtained:

$\begin{matrix}{{\delta\;{y(t)}} = {{\sum\limits_{i = 1}^{n}{{\overset{\_}{b}}_{i}\delta\;{u\left( {t - d - i} \right)}}} + {\sum\limits_{i = 1}^{n}{g_{i}\delta\;{u\left( {t - d - i^{2}} \right)}\delta\;{u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta\;{y\left( {t - i} \right)}}}}} & (20)\end{matrix}$The linear approximation for gain results in a quadratic differenceequation for the output. Given Equation 20, the approximate systematicmodel is shown to be of low computational complexity. It may be used ina MPC algorithm to efficiently compute the required control moves for atransition from one steady-state to another after a set-point change.Note that this applies to the dynamic gain variations betweensteady-state transitions and not to the actual path values.Control System Error Constraints

Referring now to FIG. 9, there is illustrated a block diagram of theprediction engine for the dynamic controller 82 of FIG. 6. Theprediction engine is operable to essentially predict a value of y(t) asthe predicted future value y(t+1). Since the prediction engine mustdetermine what the value of the output y(t) is at each future valuebetween two steady-state values, it is necessary to perform these in a“step” manner. Therefore, there will be k steps from a value of zero toa value of N, which value at k=N is the value at the “horizon”, thedesired value. This, as will be described hereinbelow, is an iterativeprocess, it being noted that the terminology for “(t+1)” refers to anincremental step, with an incremental step for the dynamic controllerbeing smaller than an incremented step for the steady-state model. Forthe steady-state model, “y(t+N)” for the dynamic model will be, “y(t+1)”for the steady state The value y(t+1) is defined as follows:y(t+1)=a ₁ y(t)+a ₂ y(t−1)+b _(i) u(t−d−1)+b ₂ u(t−d−2)  (21)

With further reference to FIG. 9, the input values u(t) for each (u,y)pair are input to a delay line 140. The output of the delay lineprovides the input value u(t) delayed by a delay value “d”. There areprovided only two operations for multiplication with the coefficients b₁and b₂, such that only two values u(t) and u(t−1) are required. Theseare both delayed and then multiplied by the coefficients b₁ and b₂ andthen input to a summing block 141. Similarly, the output value y^(p)(t)is input to a delay line 142, there being two values required formultiplication with the coefficients a₁ and a₂. The output of thismultiplication is then input to the summing block 141. The input to thedelay line 142 is either the actual input value y^(a)(t) or the iteratedoutput value of the summation block 141, which is the previous valuecomputed by the dynamic controller 82. Therefore, the summing block 141will output the predicted value y(t+1) which will then be input to amultiplexor 144. The multiplexor 144 is operable to select the actualoutput y^(a)(t) on the first operation and, thereafter, select theoutput of the summing block 141. Therefore, for a step value of k=0 thevalue y^(a)(t) will be selected by the multiplexor 144 and will belatched in a latch 145. The latch 145 will provide the predicted valuey^(p)(t+k) on an output 146. This is the predicted value of y(t) for agiven k that is input back to the input of delay line 142 formultiplication with the coefficients a₁ and a₂. This is iterated foreach value of k from k=0 to k=N.

The a₁ and a₂ values are fixed, as described above, with the b₁ and b₂values scaled. This scaling operation is performed by the coefficientmodification block 38. However, this only defines the beginningsteady-state value and the final steady-state value, with the dynamiccontroller and the optimization routines described in the presentapplication defining how the dynamic controller operates between thesteady-state values and also what the gain of the dynamic controller is.The gain specifically is what determines the modification operationperformed by the coefficient modification block 38.

In FIG. 9, the coefficients in the coefficient modification block 38 aremodified as described hereinabove with the information that is derivedfrom the steady-state model. The steady-state model is operated in acontrol application, and is comprised in part of a forward steady-statemodel 141 which is operable to receive the steady-state input valueU_(ss)(t) and predict the steady-state output value Y_(ss)(t). Thispredicted value is utilized in an inverse steady-state model 143 toreceive the desired value y^(d)(t) and the predicted output of thesteady-state model 141 and predict a future steady-state input value ormanipulated value U_(ss)(t+N) and also a future steady-state input valueY_(ss)(t+N) in addition to providing the steady-state gain K_(ss). Asdescribed hereinabove, these are utilized to generate scaled b-values.These b-values are utilized to define the gain k_(d) of the dynamicmodel. In can therefore be seen that this essentially takes a lineardynamic model with a fixed gain and allows it to have a gain thereofmodified by a non-linear model as the operating point is moved throughthe output space.

Referring now to FIG. 10, there is illustrated a block diagram of thedynamic controller and optimizer. The dynamic controller includes adynamic model 149 which basically defines the predicted value y^(p)(k)as a function of the inputs y(t), s(t) and u(t). This was essentiallythe same model that was described hereinabove with reference to FIG. 9.The model 149 predicts the output values y^(p)(k) between the twosteady-state values, as will be described hereinbelow. The model 149 ispredefined and utilizes an identification algorithm to identify the a₁,a₂, b₁ and b₂ coefficients during training. Once these are identified ina training and identification procedure, these are “fixed”. However, asdescribed hereinabove, the gain of the dynamic model is modified byscaling the coefficients b₁ and b₂. This gain scaling is not describedwith respect to the optimization operation of FIG. 10, although it canbe incorporated in the optimization operation.

The output of model 149 is input to the negative input of a summingblock 150. Summing block 150 sums the predicted output y^(p)(k) with thedesired output y^(d)(t). In effect, the desired value of y^(d)(t) iseffectively the desired steady-state value Y^(f) _(ss), although it canbe any desired value. The output of the summing block 150 comprises anerror value which is essentially the difference between the desiredvalue y^(d)(t) and the predicted value y^(p)(k). The error value ismodified by an error modification block 151, as will be describedhereinbelow, in accordance with error modification parameters in a block152. The modified error value is then input to an inverse model 153,which basically performs an optimization routine to predict a change inthe input value u(t). In effect, the optimizer 153 is utilized inconjunction with the model 149 to minimize the error output by summingblock 150. Any optimization function can be utilized, such as a MonteCarlo procedure. However, in the present invention, a gradientcalculation is utilized. In the gradient method, the gradient ∂(y)/∂(u)is calculated and then a gradient solution performed as follows:

$\begin{matrix}{{\Delta\; u_{new}} = {{\Delta\; u_{old}} + {\left( \frac{\partial(y)}{\partial(u)} \right) \times E}}} & (22)\end{matrix}$

The optimization function is performed by the inverse model 153 inaccordance with optimization constraints in a block 154. An iterationprocedure is performed with an iterate block 155 which is operable toperform an iteration with the combination of the inverse model 153 andthe predictive model 149 and output on an output line 156 the futurevalue u(t+k+1). For this will be the initial steady-state value and fork=N, this will be the value at the horizon, or at the next steady-statevalue. During the iteration procedure, the previous value of u(t+k) hasthe change value Δu added thereto. This value is utilized for that valueof k until the error is within the appropriate levels. Once it is at theappropriate level, the next u(t+k) is input to the model 149 and thevalue thereof optimized with the iterate block 155. Once the iterationprocedure is done, it is latched. As will be described hereinbelow, thisis a combination of modifying the error such that the actual erroroutput by the block 150 is not utilized by the optimizer 153 but,rather, a modified error is utilized. Alternatively, differentoptimization constraints can be utilized, which are generated by theblock 154, these being described hereinbelow.

Referring now to FIGS. 11a and 11b , there are illustrated plots of theoutput y(t+k) and the input u_(k)(t+k+1), for each k from the initialsteady-state value to the horizon steady-state value at k=N. Withspecific reference to FIG. 11a , it can be seen that the optimizationprocedure is performed utilizing multiple passes. In the first pass, theactual value u^(a)(t+k) for each k is utilized to determine the valuesof y(t+k) for each u,y pair. This is then accumulated and the valuesprocessed through the inverse model 153 and the iterate block 155 tominimize the error. This generates a new set of inputs u_(k)(t+k+1)illustrated in FIG. 11b . Therefore, the optimization after pass 1generates the values of u(t+k+1) for the second pass. In the secondpass, the values are again optimized in accordance with the variousconstraints to again generate another set of values for u(t+k+1). Thiscontinues until the overall objective function is reached. Thisobjective function is a combination of the operations as a function ofthe error and the operations as a function of the constraints, whereinthe optimization constraints may control the overall operation of theinverse model 153 or the error modification parameters in block 152 maycontrol the overall operation. Each of the optimization constraints willbe described in more detail hereinbelow.

Referring now to FIG. 12, there is illustrated a plot of y^(d)(t) andy^(p)(t). The predicted value is represented by a waveform 170 and thedesired output is represented by a waveform 172, both plotted over thehorizon between an initial steady-state value Y^(i) _(ss) and a finalsteady-state value Y^(f) _(ss). It can be seen that the desired waveformprior to k=0 is substantially equal to the predicted output. At k=0, thedesired output waveform 172 raises its level, thus creating an error. Itcan be seen that at k=0, the error is large and the system then mustadjust the manipulated variables to minimize the error and force thepredicted value to the desired value. The objective function for thecalculation of error is of the form:

$\begin{matrix}{\min\limits_{\Delta\; u_{i\; l}}{\sum\limits_{j}^{\;}{\sum\limits_{k}^{\;}\left( {A_{j}*\left( {{{\overset{\rightarrow}{y}}^{p}(t)} - {{\overset{\rightarrow}{y}}^{d}(t)}} \right)^{2}} \right.}}} & (23)\end{matrix}$where: Du_(il) is the change in input variable (IV) I at time interval l

A_(j) is the weight factor for control variable (CV) j

y^(p)(t) is the predicted value of CV j at time interval k

y^(d)(t) is the desired value of CV j.

Trajectory Weighting

The present system utilizes what is referred to as “trajectoryweighting” which encompasses the concept that one does not put aconstant degree of importance on the future predicted process behaviormatching the desired behavior at every future time set, i.e., at lowk-values. One approach could be that one is more tolerant of error inthe near term (low k-values) than farther into the future (highk-values). The basis for this logic is that the final desired behavioris more important than the path taken to arrive at the desired behavior,otherwise the path traversed would be a step function. This isillustrated in FIG. 13 wherein three possible predicted behaviors areillustrated, one represented by a curve 174 which is acceptable, one isrepresented by a different curve 176, which is also acceptable and onerepresented by a curve 178, which is unacceptable since it goes abovethe desired level on curve 172. Curves 174-178 define the desiredbehavior over the horizon for k=1 to N.

In Equation 23, the predicted curves 174-178 would be achieved byforcing the weighting factors A_(j) to be time varying. This isillustrated in FIG. 14. In FIG. 14, the weighting factor A as a functionof time is shown to have an increasing value as time and the value of kincreases. This results in the errors at the beginning of the horizon(low k-values) being weighted much less than the errors at the end ofthe horizon (high k-values). The result is more significant than merelyredistributing the weights out to the end of the control horizon at k=N.This method also adds robustness, or the ability to handle a mismatchbetween the process and the prediction model. Since the largest error isusually experienced at the beginning of the horizon, the largest changesin the independent variables will also occur at this point. If there isa mismatch between the process and the prediction (model error), theseinitial moves will be large and somewhat incorrect, which can cause poorperformance and eventually instability. By utilizing the trajectoryweighting method, the errors at the beginning of the horizon areweighted less, resulting in smaller changes in the independent variablesand, thus, more robustness.

Error Constraints

Referring now to FIG. 15, there are illustrated constraints that can beplaced upon the error. There is illustrated a predicted curve 180 and adesired curve 182, desired curve 182 essentially being a flat line. Itis desirable for the error between curve 180 and 182 to be minimized.Whenever a transient occurs at t=0, changes of some sort will berequired. It can be seen that prior to t=0, curve 182 and 180 aresubstantially the same, there being very little error between the two.However, after some type of transition, the error will increase. If arigid solution were utilized, the system would immediately respond tothis large error and attempt to reduce it in as short a time aspossible. However, a constraint frustum boundary 184 is provided whichallows the error to be large at t=0 and reduces it to a minimum level ata point 186. At point 186, this is the minimum error, which can be setto zero or to a non-zero value, corresponding to the noise level of theoutput variable to be controlled. This therefore encompasses the sameconcepts as the trajectory weighting method in that final futurebehavior is considered more important that near term behavior. The evershrinking minimum and/or maximum bounds converge from a slack positionat t=0 to the actual final desired behavior at a point 186 in theconstraint frustum method.

The difference between constraint frustums and trajectory weighting isthat constraint frustums are an absolute limit (hard constraint) whereany behavior satisfying the limit is just as acceptable as any otherbehavior that also satisfies the limit. Trajectory weighting is a methodwhere differing behaviors have graduated importance in time. It can beseen that the constraints provided by the technique of FIG. 15 requiresthat the value y^(p)(t) is prevented from exceeding the constraintvalue. Therefore, if the difference between y^(d)(t) and y^(p)(t) isgreater than that defined by the constraint boundary, then theoptimization routine will force the input values to a value that willresult in the error being less than the constraint value. In effect,this is a “clamp” on the difference between y^(p)(t) and y^(d)(t). Inthe trajectory weighting method, there is no “clamp” on the differencetherebetween; rather, there is merely an attenuation factor placed onthe error before input to the optimization network.

Trajectory weighting can be compared with other methods, there being twomethods that will be described herein, the dynamic matrix control (DMC)algorithm and the identification and command (IdCom) algorithm. The DMCalgorithm utilizes an optimization to solve the control problem byminimizing the objective function:

$\begin{matrix}{\min\limits_{\Delta\; U_{i\; l}}{\sum\limits_{j}^{\;}{\sum\limits_{k}^{\;}\left( {{A_{j}*\left( {{{\overset{\rightarrow}{y}}^{P}(t)} - {{\overset{\rightarrow}{y}}^{D}(t)}} \right)} + {\sum\limits_{i}^{\;}{B_{i}*{\sum\limits_{1}^{\;}\left( {\Delta\; U_{i\; l}} \right)^{2}}}}} \right.}}} & (24)\end{matrix}$where B_(i) is the move suppression factor for input variable I. This isdescribed in Cutler, C. R. and B. L. Ramaker, Dynamic Matrix Control—AComputer Control Algorithm, AIChE National Meeting, Houston, Tex.(April, 1979), which is incorporated herein by reference.

It is noted that the weights A_(j) and desired values y^(d)(t) areconstant for each of the control variables. As can be seen from Equation24, the optimization is a trade off between minimizing errors betweenthe control variables and their desired values and minimizing thechanges in the independent variables. Without the move suppression term,the independent variable changes resulting from the set point changeswould be quite large due to the sudden and immediate error between thepredicted and desired values. Move suppression limits the independentvariable changes, but for all circumstances, not just the initialerrors.

The IdCom algorithm utilizes a different approach. Instead of a constantdesired value, a path is defined for the control variables to take fromthe current value to the desired value. This is illustrated in FIG. 16.This path is a more gradual transition from one operation point to thenext. Nevertheless, it is still a rigidly defined path that must be met.The objective function for this algorithm takes the form:

$\begin{matrix}{\min\limits_{\Delta\; U_{i\; l}}{\sum\limits_{j}^{\;}{\sum\limits_{k}^{\;}\left( {A_{j}*\left( {Y^{P_{j\; k}} - y_{{ref}\; j\; k}} \right)} \right)^{2}}}} & (25)\end{matrix}$This technique is described in Richalet, J., A. Rault, J. L. Testud, andJ. Papon, Model

Predictive Heuristic Control: Applications to Industrial Processes,Automatica, 14, 413-428 (1978), which is incorporated herein byreference. It should be noted that the requirement of Equation 25 ateach time interval is sometimes difficult. In fact, for controlvariables that behave similarly, this can result in quite erraticindependent variable changes due to the control algorithm attempting toendlessly meet the desired path exactly.

Control algorithms such as the DMC algorithm that utilize a form ofmatrix inversion in the control calculation, cannot handle controlvariable hard constraints directly. They must treat them separately,usually in the form of a steady-state linear program. Because this isdone as a steady-state problem, the constraints are time invariant bydefinition. Moreover, since the constraints are not part of a controlcalculation, there is no protection against the controller violating thehard constraints in the transient while satisfying them at steady-state.

With further reference to FIG. 15, the boundaries at the end of theenvelope can be defined as described hereinbelow. One techniquedescribed in the prior art, W. Edwards Deming, “Out of the Crisis,”Massachusetts Institute of Technology, Center for Advanced EngineeringStudy, Cambridge Mass., Fifth Printing, September 1988, pages 327-329,describes various Monte Carlo experiments that set forth the premisethat any control actions taken to correct for common process variationactually may have a negative impact, which action may work to increasevariability rather than the desired effect of reducing variation of thecontrolled processes. Given that any process has an inherent accuracy,there should be no basis to make a change based on a difference thatlies within the accuracy limits of the system utilized to control it. Atpresent, commercial controllers fail to recognize the fact that changesare undesirable, and continually adjust the process, treating alldeviation from target, no matter how small, as a special cause deservingof control actions, i.e., they respond to even minimal changes. Overadjustment of the manipulated variables therefore will result, andincrease undesirable process variation. By placing limits on the errorwith the present filtering algorithms described herein, only controlleractions that are proven to be necessary are allowed, and thus, theprocess can settle into a reduced variation free from unmeritedcontroller disturbances. The following discussion will deal with onetechnique for doing this, this being based on statistical parameters.

Filters can be created that prevent model-based controllers from takingany action in the case where the difference between the controlledvariable measurement and the desired target value are not significant.The significance level is defined by the accuracy of the model uponwhich the controller is statistically based. This accuracy is determinedas a function of the standard deviation of the error and a predeterminedconfidence level. The confidence level is based upon the accuracy of thetraining. Since most training sets for a neural network-based model willhave “holes” therein, this will result in inaccuracies within the mappedspace. Since a neural network is an empirical model, it is only asaccurate as the training data set. Even though the model may not havebeen trained upon a given set of inputs, it will extrapolate the outputand predict a value given a set of inputs, even though these inputs aremapped across a space that is questionable. In these areas, theconfidence level in the predicted output is relatively low. This isdescribed in detail in U.S. patent application Ser. No. 08/025,184,filed Mar. 2, 1993, which is incorporated herein by reference.

Referring now to FIG. 17, there is illustrated a flowchart depicting thestatistical method for generating the filter and defining the end point186 in FIG. 15. The flowchart is initiated at a start block 200 and thenproceeds to a function block 202, wherein the control values u(t+1) arecalculated. However, prior to acquiring these control values, thefiltering operation must be a processed. The program will flow to afunction block 204 to determine the accuracy of the controller. This isdone off-line by analyzing the model predicted values compared to theactual values, and calculating the standard deviation of the error inareas where the target is undisturbed. The model accuracy of e_(m)(t) isdefined as follows:e _(m)(t)=a(t)−p(t)  (26)

where: e_(m)=model error,

-   -   a=actual value    -   p=model predicted value        The model accuracy is defined by the following equation:        Acc=H*σ _(m)  (27)

where: Acc=accuracy in terms of minimal detector error

$\begin{matrix}{H = {{{significance}\mspace{14mu}{level}} = {1\mspace{14mu} 67\%\mspace{14mu}{confidence}}}} \\{= {2\mspace{14mu} 95\%\mspace{14mu}{confidence}}} \\{= {3\mspace{14mu} 99.5\%\mspace{14mu}{confidence}}}\end{matrix}$ σ_(m) = standard  deviation  of  e_(m)(t).The program then flows to a function block 206 to compare the controllererror e_(c)(t) with the model accuracy. This is done by taking thedifference between the predicted value (measured value) and the desiredvalue. This is the controller error calculation as follows:e _(c)(t)=d(t)−m(t)  (28)

where: e_(c)=controller error

-   -   d=desired value    -   m=measured value        The program will then flow to a decision block 208 to determine        if the error is within the accuracy limits. The determination as        to whether the error is within the accuracy limits is done        utilizing Shewhart limits With this type of limit and this type        of filter, a determination is made as to whether the controller        error e_(c)(t) meets the following conditions: e_(c)(t)−1*Acc        and e_(c)(t)≦+1*Acc, then either the control action is        suppressed or not suppressed. If it is within the accuracy        limits, then the control action is suppressed and the program        flows along a “Y” path. If not, the program will flow along the        “N” path to function block 210 to accept the u(t+1) values. If        the error lies within the controller accuracy, then the program        flows along the “Y” path from decision block 208 to a function        block 212 to calculate the running accumulation of errors. This        is formed utilizing a CUSUM approach. The controller CUSUM        calculations are done as follows:        S _(low)=min(0,S _(low)(t−1)+d(t)−m(t))−Σ(m)+k)  (29)        S _(hi)=max(0,S _(hi)(t−1)+[d(t)−m(t))−Σ(m)]−k)  (30)

where: S_(hi)=Running Positive Qsum

-   -   S_(low)=Running Negative Qsum    -   k=Tuning factor—minimal detectable change threshold

with the following defined:

-   -   Hq=significance level. Values of (j,k) can be found so that the        CUSUM control chart will have significance levels equivalent to        Shewhart control charts.        The program will then flow to a decision block 214 to determine        if the CUSUM limits check out, i.e., it will determine if the        Qsum values are within the limits. If the Qsum, the accumulated        sum error, is within the established limits, the program will        then flow along the “Y” path. And, if it is not within the        limits, it will flow along the “N” path to accept the controller        values u(t+1). The limits are determined if both the value of        S_(hi)≧+1*Hq and S_(low)≦−1*Hq. Both of these actions will        result in this program flowing along the “Y” path. If it flows        along the “N” path, the sum is set equal to zero and then the        program flows to the function block 210. If the Qsum values are        within the limits, it flows along the “Y” path to a function        block 218 wherein a determination is made as to whether the user        wishes to perturb the process. If so, the program will flow        along the “Y” path to the function block 210 to accept the        control values u(t+1). If not, the program will flow along the        “N” path from decision block 218 to a function block 222 to        suppress the controller values u(t+1). The decision block 218,        when it flows along the “Y” path, is a process that allows the        user to re-identify the model for on-line adaptation, i.e.,        retrain the model. This is for the purpose of data collection        and once the data has been collected, the system is then        reactivated.

Referring now to FIG. 18, there is illustrated a block diagram of theoverall optimization procedure. In the first step of the procedure, theinitial steady-state values {Y_(ss) ^(i), U_(ss) ^(i)} and the finalsteady-state values {Y_(ss) ^(f), U_(ss) ^(f)} are determined, asdefined in blocks 226 and 228, respectively. In some calculations, boththe initial and the final steady-state values are required. The initialsteady-state values are utilized to define the coefficients a^(i), b^(i)in a block 228. As described above, this utilizes the coefficientscaling of the b-coefficients. Similarly, the steady-state values inblock 228 are utilized to define the coefficients a^(f), b^(f), it beingnoted that only the b-coefficients are also defined in a block 229. Oncethe beginning and end points are defined, it is then necessary todetermine the path therebetween. This is provided by block 230 for pathoptimization. There are two methods for determining how the dynamiccontroller traverses this path. The first, as described above, is todefine the approximate dynamic gain over the path from the initial gainto the final gain. As noted above, this can incur some instabilities.The second method is to define the input values over the horizon fromthe initial value to the final value such that the desired value Y_(ss)^(f) is achieved. Thereafter, the gain can be set for the dynamic modelby scaling the b-coefficients. As noted above, this second method doesnot necessarily force the predicted value of the output y^(p)(t) along adefined path; rather, it defines the characteristics of the model as afunction of the error between the predicted and actual values over thehorizon from the initial value to the final or desired value. Thiseffectively defines the input values for each point on the trajectoryor, alternatively, the dynamic gain along the trajectory.

Referring now to FIG. 18a , there is illustrated a diagrammaticrepresentation of the manner in which the path is mapped through theinput and output space. The steady-state model is operable to predictboth the output steady-state value Y_(ss) ^(i) at a value of k=0, theinitial steady-state value, and the output steady-state value Y_(ss)^(i) at a time t+N where k=N, the final steady-state value. At theinitial steady-state value, there is defined a region 227, which region227 comprises a surface in the output space in the proximity of theinitial steady-state value, which initial steady-state value also liesin the output space. This defines the range over which the dynamiccontroller can operate and the range over which it is valid. At thefinal steady-state value, if the gain were not changed, the dynamicmodel would not be valid. However, by utilizing the steady-state modelto calculate the steady-state gain at the final steady-state value andthen force the gain of the dynamic model to equal that of thesteady-state model, the dynamic model then becomes valid over a region229, proximate the final steady-state value. This is at a value of k=N.The problem that arises is how to define the path between the initialand final steady-state values. One possibility, as mentionedhereinabove, is to utilize the steady-state model to calculate thesteady-state gain at multiple points along the path between the initialsteady-state value and the final steady-state value and then define thedynamic gain at those points. This could be utilized in an optimizationroutine, which could require a large number of calculations. If thecomputational ability were there, this would provide a continuouscalculation for the dynamic gain along the path traversed between theinitial steady-state value and the final steady-state value utilizingthe steady-state gain. However, it is possible that the steady-statemodel is not valid in regions between the initial and final steady-statevalues, i.e., there is a low confidence level due to the fact that thetraining in those regions may not be adequate to define the modeltherein. Therefore, the dynamic gain is approximated in these regions,the primary goal being to have some adjustment of the dynamic modelalong the path between the initial and the final steady-state valuesduring the optimization procedure. This allows the dynamic operation ofthe model to be defined. This is represented by a number of surfaces 225as shown in phantom.

Referring now to FIG. 19, there is illustrated a flow chart depictingthe optimization algorithm. The program is initiated at a start block232 and then proceeds to a function block 234 to define the actual inputvalues u^(a)(t) at the beginning of the horizon, this typically beingthe steady-state value U_(ss). The program then flows to a functionblock 235 to generate the predicted values y^(p)(k) over the horizon forall k for the fixed input values. The program then flows to a functionblock 236 to generate the error E(k) over the horizon for all k for thepreviously generated y^(p)(k). These errors and the predicted values arethen accumulated, as noted by function block 238. The program then flowsto a function block 240 to optimize the value of u(t) for each value ofk in one embodiment. This will result in k-values for u(t). Of course,it is sufficient to utilize less calculations than the totalk-calculations over the horizon to provide for a more efficientalgorithm. The results of this optimization will provide the predictedchange Δu(t+k) for each value of k in a function block 242. The programthen flows to a function block 243 wherein the value of u(t+k) for eachu will be incremented by the value Δu(t+k). The program will then flowto a decision block 244 to determine if the objective function notedabove is less than or equal to a desired value. If not, the program willflow back along an “N” path to the input of function block 235 to againmake another pass. This operation was described above with respect toFIGS. 11a and 11b . When the objective function is in an acceptablelevel, the program will flow from decision block 244 along the “Y” pathto a function block 245 to set the value of u(t+k) for all u. Thisdefines the path. The program then flows to an End block 246.

Steady State Gain Determination

Referring now to FIG. 20, there is illustrated a plot of the input spaceand the error associated therewith. The input space is comprised of twovariables x₁ and x₂. The y-axis represents the function f(x₁, x₂). Inthe plane of x₁ and x₂, there is illustrated a region 250, whichrepresents the training data set. Areas outside of the region 250constitute regions of no data, i.e., a low confidence level region. Thefunction Y will have an error associated therewith. This is representedby a plane 252. However, the error in the plane 250 is only valid in aregion 254, which corresponds to the region 250. Areas outside of region254 on plane 252 have an unknown error associated therewith. As aresult, whenever the network is operated outside of the region 250 withthe error region 254, the confidence level in the network is low. Ofcourse, the confidence level will not abruptly change once outside ofthe known data regions but, rather, decreases as the distance from theknown data in the training set increases. This is represented in FIG. 21wherein the confidence is defined as α(x). It can be seen from FIG. 21that the confidence level α(x) is high in regions overlying the region250.

Once the system is operating outside of the training data regions, i.e.,in a low confidence region, the accuracy of the neural net is relativelylow. In accordance with one aspect of the preferred embodiment, a firstprinciples model g(x) is utilized to govern steady-state operation. Theswitching between the neural network model f(x) and the first principlemodels g(x) is not an abrupt switching but, rather, it is a mixture ofthe two.

The steady-state gain relationship is defined in Equation 7 and is setforth in a more simple manner as follows:

$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {f\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (31)\end{matrix}$A new output function Y(u) is defined to take into account theconfidence factor α(u) as follows:Y({right arrow over (u)})=α({right arrow over (u)})·f({right arrow over(u)})+(1−α({right arrow over (u)}))g({right arrow over (u)})  (32)

where: α(u)=confidence in model f(u)

-   -   α(u) in the range of 0→1    -   α(u)ε{0,1}        This will give rise to the relationship:

$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {Y\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (33)\end{matrix}$In calculating the steady-state gain in accordance with this Equationutilizing the output relationship Y(u), the following will result:

$\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = {{\frac{\partial\left( {\alpha\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( u^{\rightarrow} \right)} \times {F\left( \overset{\rightarrow}{u} \right)}} + {{\alpha\left( \overset{\rightarrow}{u} \right)}\frac{\partial\left( {F\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} + {\frac{\partial\left( {1 - {\alpha\left( \overset{\rightarrow}{u} \right)}} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {g\left( \overset{\rightarrow}{u} \right)}} + {\left( {1 - {\alpha\left( \overset{\rightarrow}{u} \right)}} \right)\frac{\partial\left( {g\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}}}} & (34)\end{matrix}$

Referring now to FIG. 22, there is illustrated a block diagram of theembodiment for realizing the switching between the neural network modeland the first principles model. A neural network block 300 is providedfor the function f(u), a first principle block 302 is provided for thefunction g(u) and a confidence level block 304 for the function α(u).The input u(t) is input to each of the blocks 300-304. The output ofblock 304 is processed through a subtraction block 306 to generate thefunction 1−α(u), which is input to a multiplication block 308 formultiplication with the output of the first principles block 302. Thisprovides the function (1−α(u))*g(u). Additionally, the output of theconfidence block 304 is input to a multiplication block 310 formultiplication with the output of the neural network block 300. Thisprovides the function f(u)*α(u). The output of block 308 and the outputof block 310 are input to a summation block 312 to provide the outputY(u).

Referring now to FIG. 23, there is illustrated an alternate embodimentwhich utilizes discreet switching. The output of the first principlesblock 302 and the neural network block 300 are provided and are operableto receive the input x(t). The output of the network block 300 and firstprinciples block 302 are input to a switch 320, the switch 320 operableto select either the output of the first principals block 302 or theoutput of the neural network block 300. The output of the switch 320provides the output Y(u).

The switch 320 is controlled by a domain analyzer 322. The domainanalyzer 322 is operable to receive the input x(t) and determine whetherthe domain is one that is within a valid region of the network 300. Ifnot, the switch 320 is controlled to utilize the first principlesoperation in the first principles block 302. The domain analyzer 322utilizes the training database 326 to determine the regions in which thetraining data is valid for the network 300. Alternatively, the domainanalyzer 320 could utilize the confidence factor α(u) and compare thiswith a threshold, below which the first principles model 302 would beutilized.

Although the preferred embodiment has been described in detail, itshould be understood that various changes, substitutions and alterationscan be made therein without departing from the spirit and scope of theinvention as defined by the appended claims.

Further Embodiments

For nonlinear systems, process gain may be a determining characteristicof the systems, and may vary significantly over their respectiveoperating regions. However, in systems where the process gain has littlevariance, the system may be represented by a linearization of theprocess model. In various embodiments, the linearization may be localand/or global. For example, a global linearization may be desirable insystems where the process gains do not vary significantly over theentire operation range. Alternatively, for systems that aresignificantly nonlinear over the entire operation range, a locallinearization may be used about a specific operating point, e.g., at afirst time, such as the current operating point or some other desiredoperating point. Note that the accuracy of the linearization may dependon the nonlinearity of the system and may be defined based on how muchthe operating conditions differ from those represented in the model,i.e., in the linearization. Since the linear approximation does containsome error, it is desirable from a process control standpoint to choosea well-understood linearization. For example, in one case, it may bemore desirable to have the error be small at the current operating point(at the first time) at the expense of larger error at future operatingpoints. In another case it may be desirable to have the error be smallat the desired final operating point, e.g., at a second time, at theexpense of a larger error at the current operating point. Alternatively,it may be desirable to distribute the error so that it is more or lessequal across the entire operating region. In general, the linearizationmay be any function of the gain at the current point, the desired finalpoint, and the intermediate operating points.

For example, in one embodiment, the linearization may be based on asimple or weighted average between gains at two or more operatingpoints, e.g., between the gain at the current point and the final point.In some embodiments, gains may be approximated by a linear interpolationbetween a first gain, e.g., at the current operating point, and a latergain, e.g., at the final destination point. Note that in someembodiments the linearization may be based on a single operating point,or, alternatively, a plurality of operating points, in the operatingregion.

In some embodiments, it may be desirable for model based processcontrollers to utilize process gains that are larger in magnitude thanthe actual process gain, i.e., of the actual plant or process. Morespecifically, because the size of a control move, e.g., move(s) inresponse to determined offset(s) from target operation value(s), may bebased on a multiplication of error and the inverse of the process gain,larger process gain values may be implemented in order to restrict thesize of the control move. Thus, a larger process gain will result insmaller control moves, which, in turn, cause smaller adjustments anddisruptions to the operation of the plant or process. Thus, in someembodiments, the gain may be based on the largest absolute value of thegains at one or more operating points, such as the first and the lastgain in the change of operation.

Note that the approximation methods described above regarding theprocess gains are not limited to simple linear approximations, but, infact, may be approximated via nonlinear methods as well, such as, forexample, a nonlinear interpolation between two gains in the operatingregion, e.g., the gain at the current operating point and the desiredfinal operating point.

Thus, the system may be modeled using various approximation methods,e.g., based on global and/or local linear approximations of dynamicgains based on gains at two operating points.

In some embodiments, the ratio of the gain at the current point and thedesired final operating point, e.g., at first and second times,respectively, may be used to measure the nonlinearity of the system.Because a more nonlinear system will, in general, be subject to moremodeling error, it may be desirable to configure or control otherparameters of the process control system based on the degree ofnonlinearity. In some embodiments, as indicated above, the error may beattenuated as a function of time, and/or of gain values associatedtherewith, between the current operating point and the desired finaloperating point. Alternatively, or additionally, the amount ofattenuation may be modified according to the nonlinearity of the system,e.g., a high attenuation may be used when the model is more nonlinearand correspondingly more prone to modeling error. In some embodiments, ahigher attenuation may also result in small control moves as theimportance of the control error may be reduced. Conversely, lowattenuation may be used when the model is more linear and so less proneto modeling error. In various embodiments, the attenuation may bemodified or adjusted according to various methods. For example, similarto above, the attenuation may be based on the values of gain at a firstoperating point, e.g., at the first time, and a later operating point,e.g., at the second time. Also similar to the above descriptionsregarding model approximations, the attenuation may be based on a simpleor weighted average of, a largest absolute value of, a linear ornonlinear interpolation between, and/or simply an operational pointvalue of, one or more properties of values between the current operatingpoint and the desired final operating point inclusively, e.g., the gainsof these values, among others.

Thus, the error may be attenuated according to various methods tofacilitate smoother transitions between operating points of a plant orprocess.

In some embodiments, the system may have an inherent accuracy, e.g., thedifference between the behavior of the actual system (plant or process)and the model of that system, which may be determined by comparison ofvalues of the model with actual process values. If the system ischanging dramatically in a nonlinear manner, and the corresponding modelis also changing in a nonlinear manner, it can generally be expectedthat the accuracy will be lower than if the system and model arechanging linearly.

Correspondingly, in some embodiments, the system may include an errortolerance level. For example, if the system is within a specifiedtolerance level of the desired operating point as measured from theplant or process and/or as predicted the model, no further controladjustments may be implemented, e.g., via a filter in the errorminimization device described above, because the model may not beaccurate enough to resolve small differences in the system. For a highlynonlinear system, this tolerance level may vary depending on the degreeof nonlinearity. For example, as the nonlinearity increases, the modelaccuracy may decrease, and a larger tolerance may be justified. Thus,the accuracy of the model, as determined from the gain values, maydetermine whether or not the error minimization device is used oremployed, e.g., may act as a filter regarding operation of the errorminimization device.

In one embodiment, the degree of nonlinearity and/or the tolerance maybe based on the dynamic gains at a first and second time, e.g., as afunction of the gains at these respective times. In some embodiments,the degree of nonlinearity and/or tolerance may based on a function ofthe dynamic gain k as the system moves from one operating point toanother operating point, e.g., the ratio of the gains, and/or magnitudeof the gains, among others, at the first and second times. Note that thevarious approximation and error attenuation methods described above mayalso be applied to the determination of the degree of nonlinearityand/or tolerance, e.g., via simple and weighted averages, largestabsolute values, linear and/or nonlinear interpolations, one or moreoperational point values, among others, e.g., of or between the gains atthe first and second operating point values inclusively.

Note further that each of these methods may also be applied indetermining an error constraint, e.g., an error frustum, for the model,e.g., such as those already described herein. More specifically, theerror constraint may be used for imposing a constraint on the objectivefunction, e.g., a hard and/or soft constraint. Thus, when the error isgreater than the error constraint, the constraint may be “activated” inthe objective function, i.e., a scalar factor may be applied to theconstraint as a weight for error minimization. Similar to above, theerror constraint may be based on operational point values, or propertiesthereof, at various points in time, e.g., such as the first and secondtimes described above. Thus, similar to above, the error constraint maybe based on the gains at the first and second times, e.g., via simpleand weighted averages, ratios, e.g., of the first and second respectivegains, largest absolute values, linear and/or nonlinear interpolations,one or more specific gain values, one or more magnitudes of the gains,etc.

Thus, the error of the model may be minimized via various constraints,filters, and/or other methods.

We claim:
 1. A dynamic controller, comprising: a dynamic predictivemodel for receiving a current input value from a plant or process duringoperation of the plant or process, and a desired output value, andpredicting a plurality of input values at a plurality of intermediatetimes between a first time and a second time to define a dynamicoperation path of the plant or process during operation of the plant orprocess between a current output value at the first time and the desiredoutput value at the second time; an error generator for comparing apredicted dynamic output value from the dynamic operation path to thedesired output value, and generating a primary error value as adifference between the predicted dynamic output value and the desiredoutput value at each of the plurality of intermediate times; an errorminimization device for determining a change in each input value tominimize the primary error value generated by the error generator,wherein the error minimization device determines the change in eachinput value when the error minimization device is in an operable mode,which is determined based on the primary error value and an errorconstraint during operation of the plant or process; a summation devicefor summing the determined change in each input value with an originalinput value to provide a future input value as a summed input value,wherein the original input value comprises an input value before eachintermediate time; and a controller for controlling operation of theerror minimization device to minimize the primary error value duringoperation of the plant or process in accordance with an objective of thecontroller to control operation of the plant or process.
 2. The dynamiccontroller of claim 1, wherein the error minimization device is placedinto the operable mode when the primary error value is below an errortolerance and the error minimization device is taken out of the operablemode when the primary error value is above the error tolerance.
 3. Thedynamic controller of claim 2, wherein the error tolerance is based on afunction of dynamic gain at the first time and dynamic gain at thesecond time.
 4. The dynamic controller of claim 2, wherein the errortolerance is based on a ratio of dynamic gain at the first time anddynamic gain at the second time.
 5. The dynamic controller of claim 2,wherein the error tolerance is based on a magnitude of dynamic gain atthe first time and/or dynamic gain at the second time.
 6. The dynamiccontroller of claim 2, wherein the error tolerance is based on a simpleaverage of dynamic gain at the first time and dynamic gain at the secondtime.
 7. The dynamic controller of claim 2, wherein the error toleranceis based on a weighted average of dynamic gain at the first time anddynamic gain at the second time.
 8. The dynamic controller of claim 2,wherein the error tolerance is based on a linear interpolation betweendynamic gain at the first time and dynamic gain at the second time,inclusively.
 9. A dynamic controller, comprising: a dynamic predictivemodel for receiving a current input value from a plant or process duringoperation of the plant or process, and a desired output value, andpredicting a plurality of input values at a plurality of intermediatetimes between a first time and a second time to define a dynamicoperation path of the plant or process during operation of the plant orprocess between a current output value at the first time and the desiredoutput value at the second time; an error generator for comparing apredicted dynamic output value from the dynamic operation path to thedesired output value, and generating a primary error value as adifference between the predicted dynamic output value and the desiredoutput value at each of the plurality of intermediate times; an errorminimization device for determining a change in each input value tominimize the primary error value generated by the error generator,wherein the error minimization device is operable to minimize anobjective function of the dynamic predictive model with respect to aconstraint, and wherein the error minimization device comprises an errorconstraint to determine operation of the error minimization device withrespect to the constraint of the objective function during operation ofthe plant or process; a summation device for summing the determinedchange in each input value with an original input value to provide afuture input value as a summed input value, wherein the original inputvalue comprises an input value before each intermediate time; and acontroller for controlling operation of the error minimization device tominimize the primary error value during operation of the plant orprocess in accordance with an objective of the controller to controloperation of the plant or process.
 10. The dynamic controller of claim9, wherein the error minimization device is operable to minimize theobjective function with respect to the constraint when the primary errorvalue is above the error constraint, and wherein the error minimizationdevice is not operable to minimize the objective function with respectto the constraint when the primary error value is below the errorconstraint.
 11. The dynamic controller of claim 10, wherein the errorconstraint is based on a function of dynamic gain at the first time anddynamic gain at the second time.
 12. The dynamic controller of claim 10,wherein the error constraint is based on a ratio of dynamic gain at thefirst time and dynamic gain at the second time.
 13. The dynamiccontroller of claim 10, wherein the error constraint is based on amagnitude of dynamic gain at the first time and dynamic gain at thesecond time.
 14. The dynamic controller of claim 10, wherein the errorconstraint is based on a simple average of dynamic gain at the firsttime and dynamic gain at the second time.
 15. The dynamic controller ofclaim 10, wherein the error constraint is based on a weighted average ofdynamic gain at the first time and dynamic gain at the second time. 16.The dynamic controller of claim 10, wherein the error constraint isbased on a linear interpolation between dynamic gain at the first timeand dynamic gain at the second time, inclusively.
 17. A method,comprising: receiving a current input value from a plant or processduring operation of the plant or process, and a desired output value;predicting a plurality of input values at a plurality of intermediatetimes between a first time and a second time to define a dynamicoperation path of the plant or process during operation of the plant orprocess between a current output value at the first time and the desiredoutput value at the second time; comparing a predicted dynamic outputvalue from the dynamic operation path to the desired output value;generating a primary error value as a difference between the predicteddynamic output value and the desired output value at each of theplurality of intermediate times; determining a change in each inputvalue to minimize the primary error value; summing the determined changein each input value with an original input value to provide a futureinput value as a summed input value; and controlling operation of anerror minimization device to minimize the primary error value duringoperation of the plant or process in accordance with an objective of thecontroller to control operation of the plant or process.
 18. The methodof claim 17, wherein the error minimization device is operable tominimize an objective function of a dynamic predictive model withrespect to a constraint.
 19. The method of claim 18, wherein the errorminimization device comprises an error constraint to determine operationof the error minimization device with respect to the constraint of theobjective function during operation of the plant or process.
 20. Themethod of claim 19, wherein the error minimization device is operable tominimize the objective function with respect to the constraint when theprimary error value is above the error constraint, and wherein the errorminimization device is not operable to minimize the objective functionwith respect to the constraint when the primary error value is below theerror constraint.